In a ` A B C ,` if `A B=5c m ,B C=13a n dC A=12 c m` , then the distance of vertex `' A '` from the side `B C` in (in cm) `a.(25)/(13)` b. `(60)/(13)` c. `(65)/(12)` d. `(144)/(13)`

Answer

To find the distance of vertex A from the side BC in triangle ABC, we can use the properties of right triangles and trigonometric functions. Here’s the step-by-step solution:

### Step 1: Identify the sides of the triangle
Given:
- \( AB = 5 \, \text{cm} \)
- \( BC = 13 \, \text{cm} \)
- \( CA = 12 \, \text{cm} \)

### Step 2: Confirm the triangle is a right triangle
We can check if triangle ABC is a right triangle using the Pythagorean theorem:
\[ AB^2 + CA^2 = BC^2 \]
Calculating:
\[ 5^2 + 12^2 = 25 + 144 = 169 \]
\[ BC^2 = 13^2 = 169 \]
Since both sides are equal, triangle ABC is a right triangle with the right angle at A.

### Step 3: Use the sine function to find the height from A to side BC
In a right triangle, the height (perpendicular distance from vertex A to side BC) can be calculated using the sine of angle A:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Here, the opposite side is the height (let's denote it as \( h \)), and the hypotenuse is \( AB \):
\[ \sin(\theta) = \frac{h}{AB} \]
We can find \( \sin(\theta) \) using the sides of the triangle:
\[ \sin(\theta) = \frac{CA}{BC} = \frac{12}{13} \]

### Step 4: Substitute to find the height
Now substituting into the sine equation:
\[ \frac{12}{13} = \frac{h}{5} \]
Cross-multiplying gives:
\[ 12 \cdot 5 = 13h \]
\[ 60 = 13h \]
Now, solving for \( h \):
\[ h = \frac{60}{13} \]

### Conclusion
The distance of vertex A from the side BC is:
\[ \boxed{\frac{60}{13} \, \text{cm}} \]